Quasi‐symmetric designs with fixed difference of block intersection numbers

The following results for proper quasi‐symmetric designs with non‐zero intersection numbers x , y and λ > 1 are proved. (1) Let D be a quasi‐symmetric design with z  =  y  −  x and v  ≥ 2 k . If x  ≥ 1 +  z  +  z 3 then λ <  x  + 1 +  z  +  z 3 . (2) Let D be a quasi‐symmetric design with intersection numbers x , y and y  −  x  = 1. Then D is a design with parameters v  = (1 +  m ) (2 +  m )/2, b  = (2 +  m ) (3 +  m )/2, r  =  m  + 3, k  =  m  + 1, λ = 2, x  = 1, y  = 2 and m  = 2,3,… or complement of one of these design or D is a design with parameters v  = 5, b  = 10, r  = 6, k  = 3, λ = 3, and x  = 1, y  = 2. (3) Let D be a triangle free quasi‐symmetric design with z  =  y  −  x and v  ≥ 2 k , then x  ≤  z  +  z 2 . (4) For fixed z  ≥ 1 there exist finitely many triangle free quasi‐symmetric designs non‐zero intersection numbers x , y  =  x  +  z . (5) There do not exist triangle free quasi‐symmetric designs with non‐zero intersection numbers x , y  =  x  + 2. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 49–60, 2007